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Generalized divisor function. Number of partitions of n with exactly three part sizes.
(Formerly M1367 N0530)
9

%I M1367 N0530 #41 Sep 15 2023 18:45:54

%S 1,2,5,10,15,25,37,52,67,97,117,154,184,235,277,338,385,469,531,630,

%T 698,810,910,1038,1144,1295,1425,1577,1741,1938,2089,2301,2505,2700,

%U 2970,3189,3444,3703,4004,4242,4617,4882,5244,5558,5999,6221,6755,7050,7576

%N Generalized divisor function. Number of partitions of n with exactly three part sizes.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A002134/b002134.txt">Table of n, a(n) for n = 6..10000</a>

%H P. A. MacMahon, <a href="https://doi.org/10.1112/plms/s2-19.1.75">Divisors of numbers and their continuations in the theory of partitions</a>, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

%F G.f.: Sum_{i>=1} Sum_{j=1..i-1} Sum_{k=1..j-1} x^(i+j+k)/((1-x^i)*(1-x^j)* (1-x^k)). - _Geoffrey Critzer_, Sep 13 2012

%e a(8) = 5 because we have 5+2+1, 4+3+1, 4+2+1+1, 3+2+2+1, 3+2+1+1+1.

%p # Using function P from A365676:

%p A002134 := n -> P(n, 3, n): seq(A002134(n), n = 6..54); # _Peter Luschny_, Sep 15 2023

%t nn=40;sss=Sum[Sum[Sum[x^(i+j+k)/(1-x^i)/(1-x^j)/(1-x^k),{k,1,j-1}], {j,1,i-1}], {i,1,nn}]; Drop[CoefficientList[Series[sss,{x,0,nn}],x],6] (* _Geoffrey Critzer_, Sep 13 2012 *)

%Y A diagonal of A060177.

%Y Column k=3 of A116608. - _Alois P. Heinz_, Nov 07 2012

%K nonn,easy

%O 6,2

%A _N. J. A. Sloane_

%E Better description and more terms from _Naohiro Nomoto_, Jan 24 2002

%E More terms from _Vladeta Jovovic_, Nov 02 2003